Gas Turbines Course Book v2[1][3][1].0 14 March 06
Short Description
Introduccion a la termodinamica, seguido de de la teoria de turbians a gas....
Description
Gas turbines
Gas Turbines, WB4420 / 4421 Faculty of Mechanical, Maritime and Materials Engineering, TU Delft
Thermodynamics and Gas Turbines, AE3–235 Faculty of Aerospace Engineering, TU Delft
Editors-in-Chief:
Prof. Ir. J.P. van Buijtenen Chair of Gas Turbines, Delft University of Technology, The Netherlands and
Ir. Wilfried Visser Manager, Delta Consult, The Netherlands
1
Gas turbines
Authors: Prof. Ir. Jos P. van Buijtenen, Chair of Gas Turbines, TU Delft (Introduction, Ideal Cycles, Real Cycles, Shaft power Gas turbines, Turbo machinery)
Ir.
Wilfried
P.J.
Visser,
Manager,
Delta
Consult,
Ex-NLR
Scientist
(Introduction, Ideal Cycles, Real Cycles, Shaft power Gas turbines, Aircraft Gas Turbines and Performance Characteristics)
Ir. Tiedo Tinga, Scientist, National Aerospace Laboratory (NLR) (Loads and Materials)
Savad Shakariyants, M.Sc, Energy Technology, TU Delft (Combustion Chamber)
Francesco Montella, M.Sc, Energy Technology, TU Delft (Turbomachinery)
Compiled by: Jitendra Singh, B.E.(Hons.) (Ex Engineer-General Electric Company, GE Global Research) Aerospace Engineering - Masters student, TU Delft.
Date of Revision: 10 March 2006. Second Edition
© All rights reserved. No part of this book may be reproduced and/or disclosed, in any form or by any means without the prior written permission of the owners.
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Gas turbines
Contents
1
2
3
Introduction
7
1.1 The gas turbine engine concept
7
1.2 History
10
1.2.1
The first industrial gas turbines
10
1.2.2
The first jet engines
11
1.2.3
Gas turbine research and development
12
1.3 Application areas
13
1.4 Gas turbine engine manufacturers
13
1.5 Performance
14
1.6 Gas turbine configurations
15
Ideal cycles
17
2.1 The Joule-Brayton cycle
17
2.2 Performance analysis of an ideal simple cycle
19
2.3 Example
23
2.4 Enhanced cycles
26
2.4.1
Heat exchange
26
2.4.2
Intercooling
30
2.4.3
Reheat
33
2.4.4
Combined intercooling, reheat and recuperation
36
Real cycles
38
3.1 Deviations with respect to the ideal process
38
3.2 Specific heat cp and specific heat ratio k
40
3.3 Total enthalpy, temperature and pressure
41
3.4 Compressor and turbine efficiency
42
3.5 Pressure losses
47
3.5.1
Combustion chamber pressure loss
47
3.5.2
Inlet pressure losses in industrial gas turbines
47
3.5.3
Inlet pressure losses in aircraft gas turbines
48
3.5.4
Exhaust system pressure losses in industrial gas turbines
48
3.5.5
Exhaust system pressure losses in aircraft gas turbines
49
3.6 Mechanical losses
49
3.7 Combustor efficiency
49
3.8 Calculation scheme to determine gas generator power and efficiency
49
3.9 Performance characteristics of the gas generator
51 3
Gas turbines
4
5
6
3.10 Example: Real gas generator
54
3.11 Real enhanced cycles
56
3.11.1
Recuperated cycles and heat exchanger effectiveness
56
3.11.2
Combined intercooling and heat exchange
57
3.11.3
Reheated cycles
58
Shaft power gas turbines
60
4.1 Introduction
60
4.2 Single or multi spool configurations
60
4.3 Specific power and thermal efficiency as function of the process parameters
61
4.4 Enhanced cycles
64
4.4.1
Recuperators and regenerators
64
4.4.2
Intercooling
64
4.4.3
Reheat
64
4.5 Using exhaust gas waste heat
64
4.5.1
Configurations
64
4.5.2
Effects of system parameters on cycle performance
66
Aircraft gas turbines
69
5.1 Aircraft propulsion
69
5.2 Thrust equation
69
5.3 Determining thrust
70
5.4 Installed and uninstalled thrust
72
5.5 Propulsion system power and efficiencies
74
Combustion
76
6.1 Introduction
77
6.2 Fuels
78
6.3 Heat Release
80
6.4 Simplified Combustor Heat Balance
88
6.5 Combustor Components
92
6.6 Flame Stabilization
97
6.7 Cooling
98
6.8 Combustor Types
100
6.9 Flow Direction
102
6.10 Combustion Performance
102
6.10.1
Ignition
102
6.10.2
Combustion Stability
103
6.10.3
Heat Losses and Incomplete Combustion
105 4
Gas turbines
7
8
9
6.11 Pollutant Emission
108
Turbomachinery
118
7.1 History
118
7.2 Change of Velocities in a turbo-machine
119
7.3 Euler’s Equation
120
7.4 The Axial Compressor
122
7.5 The Radial Compressor
127
7.6 The Axial Turbine
128
7.7 Characteristic Performance of a Compressor
129
Performance characteristics
131
8.1 Component characteristics
131
8.1.1
Dimensionless parameter groups
131
8.1.2
Operational limits
134
8.2 Gas turbine system characteristics
140
8.2.1
Gas generator characteristics
140
8.2.2
System characteristics of different applications
141
Loads and materials
145
9.1 Loads
145
9.1.1 Centrifugal loads
145
9.1.2 Thermal loads
145
9.1.3 Vibration loads
146
9.1.4 Pressure loads
146
9.2 Design Criteria
147
9.2.1 Static strength
147
9.2 .2 Fatigue
148
9.2.3 Creep
153
9.2.4 Oxidation and corrosion
155
9.2.5 Design criteria overview
156
9.3 Materials
156
9.3.1 Compressor blades
157
9.3.2 Combustion chamber
158
9.3.3 Turbine rotor blades
158
9.3.4 Turbine stator vanes
161
9.3.5 Turbine and compressor discs
161
9.3.6 Summary
163
9.4 Manufacturing aspects
164 5
Gas turbines
9.4.1 Casting
164
9.4.2 Coatings
165
9.5 Structural design philosophies
167
9.5.1 Safe-Life
167
9.5.2 Damage Tolerance
168
9.5.3 Retirement for Cause
169
9.5.4 Application to gas turbines
169
Appendix A
Station numbering
172
Appendix B
Acronyms
175
Appendix C
Glossary
176
Appendix D
Suggested Readings
179
6
Gas turbines
1
Introduction
(Prof. Ir. Jos P. van Buijtenen, Ir. Wilfried P.J. Visser)
1.1 The gas turbine engine concept The gas turbine engine is a machine delivering mechanical power (or thrust in case of a jet engine) using a gaseous working fluid. It is an internal combustion engine like the reciprocating Otto- and Diesel piston engines with the major difference that the working fluid flows through the gas turbine continuously and not intermittently. The continuous flow of the working fluid requires the compression, heat input, and expansion to take place in separate components. For that reason a gas turbine consists of at least a compressor, a combustion chamber and a turbine. Even though a gas turbine engine consists of more components than just a turbine, it is named after that single component. This is for historical reasons because the gas turbine was developed as an alternative for the steam turbine. The compression component of a steam cycle, the water pump, usually receives far less attention than the gas expansion component (i.e. the turbine). More obvious designations for the gas turbine and its components would be turbo compressor, and turbo expander for respectively the compression- and the expansion part and turbo engine for the whole engine.
Figure 1.1 - Alstom Typhoon (previously Ruston) 4900 kW single shaft gas turbine for generator drive Figure 1.1 shows a gas turbine delivering shaft power, consisting of a single compressor, combustion chamber and turbine. Figure 1.2 shows a “turbofan” jet engine used for aircraft propulsion.
7
Gas turbines
Figure 1.2 - IAE V2500 turbofan engine (application: Airbus A320 and other aircraft) Gas turbine configurations may differ due to the use of different types of components. There are both axial and radial compressors and turbines referring to the main direction of flow inside the component. In axial components the airflow flows axially (parallel to the rotor drive shaft) through the component, while in radial components the flow is diverted from an axial to a radial direction in case of compressor components, and vice versa for the turbine components. Also, combustion chambers come in various types: multiple small combustion chambers or annular type combustion chambers for example (Figure 1.6). The different types of compressors, turbines, and combustion chambers will be discussed in more detail in the following chapters.
low pressure power turbine high pressure turbine compressor
5 exhaust
combustor
gas generator g Figure 1.3 - Free power turbine configuration The free power turbine in Figure 1.3 converts the potential energy of the gas generator exhaust gas into mechanical work. The shaft of the free power turbine can be used to drive a car, a
8
Gas turbines
pump, a propeller (aircraft or ship), or a helicopter rotor (Figure 1.4). The high-pressure gas can also be converted into kinetic energy by expansion in a nozzle or jet pipe for aircraft propulsion (Figure 1.6). The various power conversion processes will be further addressed in the following chapters.
Figure 1.4 - Allison C250 485 kW free power turbine configuration for helicopter propulsion (Bo107/115 helicopter)
Figure 1.5 - Longitudinal cross-section of Allison C250 gas turbine
9
Gas turbines
Figure 1.6 - General Electric J-85 turbojet engine 1.2 History The history of the gas turbine is, when compared to the steam turbine and the Otto- and Diesel piston engines, relatively young. The first (usable) steam turbines were already built during the second half of the 19th century by De Laval, Parsons, and Curtis and others. The first practically useful gas turbine engines emerged at the beginning of the 20th century but largescale application only started after WWII. The reason is the specific nature of the gas turbine thermodynamic process. All gas or steam cycle processes, produce useful power only if the power required for compression is less than the power delivered by expansion. In a steam cycle the compression power of the feed water is relatively low and losses do not play a significant role. The highest process (steam) temperature is limited, but when using a condenser the pressure ratio for expansion of the steam is high. The compression power of the gas turbine cycle however, is relatively high. For the expansion of the gas, a pressure ratio equal to the compression pressure ratio minus some pressure losses is available. This means any surplus turbine power (the difference between compression and expansion power) can only be the result of the higher temperature level (compared to compressor entry temperature) at the start of the expansion in the turbine. Gas turbine compression power typically is 2/3rd of the expansion power used for driving the compressor. This means useful power is the difference between two large values and this makes losses in the compression- and expansion processes very significant for overall efficiency. 1.2.1
The first industrial gas turbines
The first experimental gas turbine engines were not able to run self-sustained, but required an external power source. Only in 1905, the Frenchman Rateau built a gas turbine that actually delivered shaft power with 25 centrifugal compressor stages delivering a pressure ratio of 3. This pressure ratio would normally not suffice for a gas turbine to deliver power, but with an extremely high combustion temperature combined with water-cooled turbine blades, Rateau managed to generate some useful power. However, the thermal efficiency of this gas turbine was only 3.5%. Further development of the gas turbine continued, especially in Switzerland by
10
Gas turbines
Prof. Stodola of the University of Zurich and manufacturer Brown Boveri (currently named ABB). Brown Boveri pioneered in the development of gas turbines for electrical power generation and other industrial applications. The first gas turbine for power generation became operational in 1939 in Neufchateau, Switzerland (Figure 1.7).
Figure 1.7 - Brown-Boveri industrial 4 MW gas turbine in 1939 The gas turbines of the early years were mainly used to provide power at peak loads. This is because the gas turbine can start up relatively quickly, requires relatively low investment costs and short production times. The low thermal efficiency as compared to steam turbines is of less concern due to the relatively small number of peak load operating hours. Only during the 1980’s, the gas turbine had its breakthrough in the power generation application. This happened due to the availability of natural gas as a fuel, which made the gas turbine particularly attractive for integration in existing natural gas fired power stations into a combined cycle unit. Also in cogeneration installations for industries consuming large amounts of heat, the gas turbine became very popular. 1.2.2
The first jet engines
In the same period that the gas turbine developed for power generation and industrial applications, Frank Whittle (England), Hans von Ohain, Herbert Wagner, and Helmut Schelp (Germany) independently started the development of a jet engine gas turbine for aircraft propulsion. Frank Whittle, at that time flying officer in the Royal Air Force, first considers the concept of the gas turbine as a jet engine in 1929 and is the first to claim a patent on the concept in 1930. Whittle set a target to design an aircraft engine capable of operating at altitudes and speeds (up
11
Gas turbines
to 900 km/h), which were far beyond the operating limits of piston engines and propellers. The British government as well as the British aircraft engine manufacturers did not share Whittles enthusiasm and did not support Whittle financially nor technically. In 1936 Whittle and some friends and investors establish a company called “Power Jets Limited”. In spite of many technological problems and a lack of funds he eventually builds his first gas turbine. During the late 30’s, Whittle draws attention with an engine running on a test bed and suddenly gets financial support from the British government. Now Whittle is able to rapidly solve technological difficulties and finally builds his first jet engine for the Gloster E28 in the year 1941. This successful achievement results in further development of Whittles jet engine design by others (Rover, Rolls Royce and General Electric). The first operational British jet fighter, the Gloster Meteor, flies in August 1944 and is initially used for interception of German V-1 missiles. Although Frank Whittle was the first to register a patent for the jet engine concept, it was Hans von Ohain who first built a gas turbine in a jet engine configuration. After completion of his study in physics in 1936, Von Ohain started to work for aircraft constructor Ernst Heinkel. Due to Heinkel’s desire to build the world’s fastest aircraft, Von Ohain receives the substantial support needed to develop a jet engine. In 1937, Von Ohain designs a simple gas turbine with a radial compressor, a combustor running on hydrogen and a radial turbine. After a number of successful tests, Von Ohain received more support from Heinkel, enabling him to demonstrate the historic first flight of the jet engine powered Heinkel He-178 aircraft in 1939. Von Ohain not only proved the concept of jet propulsion but also proved that with a jet engine, very favorable thrust-to-weight ratios can be achieved when compared to piston engines with propellers. In Germany, also Herbert Wagner and Helmut Schelp worked on the development of gas turbine jet engines. Helmut Schelp contributed to the development of the successful and first operational Messerschmidt Me-262 jet fighter. Helmut Wagner worked for Junkers on a gas turbine driving a propeller. 1.2.3
Gas turbine research and development
After the WWII, the gas turbine rapidly develops towards a powerful new alternative for industrial and aircraft applications. The development of high-temperature materials and later also cooling techniques enables the gas turbine to operate at higher turbine inlet temperatures. Extensive research in the aerodynamics improves the efficiencies of compressors and turbines. With the development of new gas turbine configurations (e.g. turbofan aircraft engines and combined-cycle concepts for stationary applications), which further improved performance and efficiency, it has become the primary choice for many applications.
12
Gas turbines
Currently, gas turbine research and development is focused on many different disciplines. The most important ones are: •
Aerodynamics:
compressor and turbine stage efficiency and loading, cooling, clearance control, noise, etc.
•
Materials:
high-temperature alloys, strength, life, coatings, and ceramics.
•
Combustion:
high-efficient, stable, low-emission combustion in short and small combustors.
•
System performance: cycle optimization, combined cycle concepts.
1.3 Application areas In section 1.1 the concept of the gas turbine has been explained of a gas generator providing hot, high-pressure gas. The way the energy in the hot gas (i.e. the ‘gas power’) is used depends on the application. This means that in general, the gas generator may be considered a subsystem that all gas turbine engines have in common while the systems converting the gas power can be very different. Although all gas generators have the same function and most will have the same configuration, significant differences exist also for the gas generator depending on the applications. These usually result from requirements with respect to •
Power output (ranging from several tens of megawatts for the larger aircraft gas turbines to several hundreds of megawatts for large power generation heavy-duty gas turbines)
•
Volume and weight (e.g. for aerospace applications).
•
Operating profile (e.g. electricity base load generation with almost constant operating conditions and power setting or the usually large variations in power setting in a helicopter or a fighter aircraft).
•
Fuel type.
•
Emissions of pollutant exhaust gasses and noise.
•
Operating conditions (corrosion, erosion), etc.
The diversity in requirements and consequences for the design has led to a division into separate groups of gas turbine manufacturers for aircraft gas turbines and industrial gas turbines. 1.4 Gas turbine engine manufacturers The largest manufacturer for industrial gas turbines at the moment is General Electric – USA (GE). GE’s share of the market is 70 percent. The other manufacturers share the remaining part of the market; among them are Alstom (several European countries, includes former Asea Brown Boveri ABB, Alsthom, European Gas Turbines), Siemens from Germany (includes KWU and Westinghouse from USA), Mitsubishi Heavy Industries in Japan and several other small manufacturers. World wide, about 1000 industrial gas turbines are sold annually. GE is also the largest manufacturer of aircraft gas turbines, followed by Rolls Royce (UK, includes Allison), Pratt & Whitney (USA/Canada), Honeywell (USA, includes Allied Signal
13
Gas turbines
and Garret), Snecma (France, includes Turbomeca), MTU (Germany), FiatAvio (Italy), Japanese Aero Engine Corporation (JAEC), and some other small manufacturers. The costs and also the risks of R&D for new advanced gas turbines are very high and have forced many manufacturers to collaborate with other manufacturers. Sometimes a manufacturer develops a new engine, and other companies develop one or more modules. Sometime joint ventures are established with several partners and engines are designed and produced under the new joint venture name. Examples of collaborations are: •
CFM (GE and Snecma, CFM-56 engine),
•
GE with Snecma, IHI and FiatAvio (GE90 turbofan engine for the B777),
•
IAE (International Aero Engines, Rolls-Royce, Pratt & Whitney (USA), JAEC, FiatAvio and MTU united in 1983 to develop the IAE-V2500 engine, see Figure 1.2),
•
Turbo-Union (Rolls-Royce, FiatAvio and MTU (RB199 for the Panavia Tornado),
•
BWM-RR (Rolls Royce and BMW (regional and business jet BR700 series engines).
The Russian industrial and aircraft gas turbine industry is significant in size, but, since the end of the Soviet Union is still struggling to become competitive with the other manufacturers. 1.5 Performance Aircraft gas turbines are manufactured in a wide thrust range. From small gas turbines for remotely piloted aircraft with 40 to 100 Newtons of thrust up to about 400 kN (Rolls-Royce Trent, GE90). Industrial gas turbines range from 200 kW (Kawasaki) up to 240 MW (ABB). Several aircraft gas turbine designs have derivatives for stationary applications on the ground. These usually are referred to as ‘aeroderived’ industrial gas turbines. Examples are the aeroderived versions of the Rolls-Royce Avon, Spey, Olympus, RB211 and Trent engines. The GE LM2500 and LM6000 industrial gas turbines are ‘aeroderivatives’ of the CF6-50 and CF680 engines respectively.
Figure 1.8 - Rolls-Royce Trent turbofan (top) and ‘aeroderived’ turboshaft (bottom) 14
Gas turbines
If the large fan at the front and the exhaust nozzle at the end of the turbofan in Figure 1.8 would be removed, a gas generator or ‘core engine’ remains capable of providing gas power applications other than providing thrust to an aircraft. The lower half of Figure 1.8 is an image of the ‘aeroderived’ industrial version of the RB211 engine: with a suitable inlet and the lowpressure turbine is coupled to a drive shaft, a turboshaft engine is created for delivering shaft power. The low-pressure turbine, which originally drove the fan that consumed most of the available power for generating thrust, now is used for proving shaft power. The removal of the fan, which also contributes to the compression of the gas generator, results in a small decrease in overall compression ratio. The low-pressure speed often is in the range suitable for generator drive (3000/3600 rpm for 50/60 Hz electrical AC power). For jet engines, power output generally is specified in terms of thrust (kN of lbs). To compare with shaft power output, jet engine thrust can be multiplied with aircraft air speed to obtain ‘propulsion power’. In chapter 5 the issues with jet engine performance in will be further addressed.
1.6 Gas turbine configurations In the previous sections it was explained that the configuration of the gas turbine is highly dependent on the type of application. Figure 1.9 and Figure 1.10 show some common turboshaft configurations for providing shaft power. Figure 1.11 and Figure 1.12 show some jet engine configurations.
Figure 1.9 -.Single-spool turboshaft
Figure 1.10 - Twin-spool turboshaft
Single-spool gas generator with free power turbine
Twin-spool turboshaft with free power turbine 15
Gas turbines
Figure 1.11 - Single-spool turbojet
Twin-spool turbojet
Figure 1.12 - Twin-spool turbofan
Twin-spool mixed turbofan
16
Gas turbines
2
Ideal cycles
(Prof. Ir. Jos P. van Buijtenen, Ir. Wilfried P.J. Visser)
2.1 The Joule-Brayton cycle The Joule-Brayton cycle represents the thermodynamic process in the gas turbine. Apart from the continuous flow of the medium through the gas turbine (see the previous chapter), another distinctive property of the Joule-Brayton cycle is that heat input (usually combustion) is taking place at constant pressure rather than at constant volume, as is the case with a piston engine. Also, the cycle can either be open or closed. In an open cycle, atmospheric air is drawn into the gas turbine compressor continuously and heat is added, usually by the combustion of fuel. The hot combustion gas is expanded in a turbine and ejected into the atmosphere, as shown in Figure 2.1(a). In a closed cycle, the same working fluid, be it air or some other gas, is circulated through the gas turbine and heat is usually added by a heat exchanger, as shown in Figure 2.1(b). An open or closed cycle gas turbine process, as depicted in Figure 2.1(a) and (b), would ideally be represented by the cycle depicted in Figure 2.2. Ignoring irreversibility, meaning ignoring pressure drops due to friction and heat losses to the surroundings, the ideal cycle is composed of two isentropic (lines 2-3 and 4-5) and two isobaric (lines 2-3 and 4-1) processes. The cycle resulting from these idealizations is called the Joule (or Brayton) cycle, often also referred to as ideal simple cycle.
Gas Generator inlet air
1
exhaust 2
3
4
air
g 5
heat input power extraction
compression
expansion
heat extraction air or other gas
open cycle (a) closed cycle (b)
Figure 2.1 – Open and Closed Cycle
17
Gas turbines
p = constant
h 4
3
g 5
2
s Figure 2.2 - The ideal gas turbine cycle h-s (enthalpy – entropy) diagram With respect to the real gas turbine process, the ideal cycle assumes the following simplifications: 1. The ideal cycle’s working fluid is considered an ideal gas having constant specific heats Cp &Cv and constant composition. For numerical calculations, values for specific heat Cp and specific heat ratio k are obtained from air at atmospheric conditions. Because of the “ideal” air working fluid the cycle is called the “ideal air cycle”. 2. Changes in kinetic and potential energy between inlet and exit of the various components can be ignored. 3. The compression and expansion processes are isentropic (i.e. reversible and adiabatic). 4. In a closed cycle, there is heat transfer during transition 5-2 (see Fig 2.2) to arrive at condition 2. In an open cycle, the atmosphere can be considered as a heat exchanger that cools down the exhaust gases at the inlet pressure (see 2.1(a). Both processes can be modeled using the same cycle in Fig 2.2 5. Pressure losses in the heat exchanger 3-4 (the combustion chamber), in the heat exchanger 5-2, in the connections between the components, in the in- and exit are ignored. 6. Constant mass flow rate of the circulating medium 7. Mechanical losses with transmission of expansion power to the compression process are ignored. Between stations 4 and 5 (i.e. the expansion process), station g can be identified in the h-s diagram (see fig. 2.2). The position of this point is such that the distance 4-g equals distance 2-3, representing the required specific compression power. The process 2-3-4-g represents the process that takes place in the gas generator. The residual power, represented by g-5, is the 18
Gas turbines
specific gas power. Gas power is defined as the power that can be extracted from the hot pressurized gas with 100% isentropic efficiency (i.e. the maximum mechanical shaft or thrust power that would be obtained under ideal conditions with an ideal 100% efficiency turbine). Specific gas power is gas power per unit of mass flow. With the above-defined simplifications, the cycle variable parameters are ambient conditions p2 and T2, end-compression pressure p3, maximum cycle temperature T4 and mass flow. 2.2 Performance analysis of an ideal simple cycle In this section the physical relations of the cycle parameters with specific gas power and efficiency are explained. These relations indicate how an ideal cycle can be optimized in terms of power output and efficiency. For a real cycle, the cycle relations show significant deviations from the ideal cycle, but they still roughly point in the same direction. Therefore, for a preliminary assessment of gas turbine cycle configurations, analysis of the ideal cycle equations provides valuable information. The exchange of mechanical power and heat among the various components of the ideal cycle gas turbine can be calculated using the following equations: Compressor power:
W 2−3 = m c p (T3 − T2 )
[W ]
(2.1)
[W ]
(2.2)
[W ]
(2.3)
Heat input rate:
Q 3−4 = m c p (T4 − T3 )
Turbine power:
W 4− g = m c p (T4 − Tg )
Gas power:
W gg = W g −5 = m c p (Tg − T5 )
(2.4)
Waste heat:
Q 5−2 = m c p (T5 − T2 )
(2.5)
Ideal (isentropic) gas equation: k
p3 T3 k −1 = p2 T2
(2.6)
19
Gas turbines
Since the compression and the expansion are isentropic and k is constant, the pressure ratio of the compression process (2-3) equals the pressure ratio of the expansion process (4-5): k
k
p p T k −1 T4 k −1 ε = 3 = 4 = 3 = p2 p5 T2 T5
(2.7)
Also applicable for g-4
pg
k
Tg k −1 = p4 T4
(2.8)
The obtained work of 4-g equals the work of 2-3,W4-g = W2-3, meaning Tg = T4 – T3 + T2. Using equation (2.7):
(
Tg = T4 − T2 ε
k −1 k
)
−1
(2.9)
Using equation (2.8) it follows: k
k
k −1 T k −1 T k −1 pg = p3 g = p2 ε 1 − 2 ε k − 1 T4 T4
(
)
(2.10)
Substituting equation (2.7) and (2.9)into equation (2.4), and dividing the gas power Wgg by the mass flow, the specific gas power is obtained: k −1 1 Ws , gg = c p (Tg − T5 ) = c p T4 1 − k −1 − c p T2 ε k − 1 εk
(2.11)
In dimensionless form:
k −1 W 1 k s, gg T4 = 1− −1 − ε c T T k −1 p2 2 ε k
(2.12)
Specific gas power can be used as a measure for the compactness of the gas generator (i.e. diameter). Gas generator dimensions together with maximum power output are important properties for the gas turbine application type. A large specific gas power means a relatively small mass flow and for a certain flow velocity (because of m=¼πρD2) a relatively small flow passage. The relation between specific gas power and volume or weight of the gas generator is more complex. The length of the gas generator is determined by pressure ratio ε and compressor technology level (pressure ratio achieved per compressor stage). For a certain stage pressure ratio, the number of compressor stages increases with cycle pressure ratio. For the turbine, this
20
Gas turbines
relation is less severe since turbine stage pressure ratios do not suffer from aerodynamic limitations as the compressor does (see chapter 7 on turbomachinery). Thermodynamic efficiency is defined as the ratio of gas power over heat added to the process:
ηtherm.dyn. =
Ws , gg Qs ,3− 4
Tg − T5
=
(2.13)
T4 − T3
Substituting Tg from equation (2.9)and T2 and T4 from (2.7) the following equation is obtained:
η therm.dyn.
T2 1 = 1 − = 1 − κ −1 T3 ε κ
(2.14)
Ideal cycle thermodynamic efficiency only depends on pressure ratio ε and specific heat ratio k. k depends on the type and temperature of the fluid used in the cycle; in a gas turbine usually air. In simplified calculations and also in this text book k is considered a constant in the equations derived above. Figure 2.3 shows the relation between the specific gas power and the thermodynamic efficiency as function of the temperature ratio T4 /T2 and the pressure ratio ε (equation (2.12) and (2.14). The figure shows there is a trade off between lower pressure ratio (with benefits in terms of low weight and small volume) and higher-pressure ratio (high thermal efficiency, i.e. low specific fuel consumption).Figure 2.3 - Ideal cycle performance
64
0,7
ηthermodyn
32
ε opt
0,6
16
0,5 8 0,4 4
0,3 0,2 3 0,1
4 5 6 7
T4
0 0
0,5
ε
2
T2 1,0
1,5
2,0 2,5 W s, gg
3,0
cp T2
Figure 2.3 - Ideal cycle performance The peak value of specific power for a given temperature ratio T4 /T2 is called the optimum pressure ratio, εopt (see the dashed curve in Figure 2.3). One way to obtain the optimum pressure
21
Gas turbines
ratio is to differentiate the equation (2.12) using the ε as variable. Another method is to differentiate equation (2.4) using T3 (which has a direct relation with ε via equation (2.6) as a variable as follows:
Ws , gg = c p (Tg − T5 ) = c p [(T4 − T5 ) − (T3 − T2 )]
[W / kg / s ]
(2.15)
Since the following equation holds from the isentropic gas equation
ε
k −1 k
=
T3 T4 = T2 T5
then T5 =
T4 T2 T3
(2.16)
equation (2.15) can be written to
TT Ws , gg = c p T4 − 4 2 − T3 + T2 T3
(2.17)
Differentiate equation (2.17) using T3 as variable for a given T2 and T4 , the equation becomes: d dT3
T T Ws , gg = 0 ⇒ c p 4 22 − 1 ⇒ T32 = T2T4 T3
(2.18)
Thus, T3 for maximum gas power is:
T3 = T2T4
(2.19)
Then εopt can be written as: k
ε opt
T k −1 T T = 3 = 2 4 T2 T2
k
k
k −1 T4 2( k −1) = T2
(2.20)
Using equation 2.16 and 2.19, at the optimum pressure ratio the following result is obtained:
T3 = T5
(2.21)
The specific power and the thermodynamic efficiency for the optimum pressure ratio are respectively:
Ws , gg c pT2
T = 4 − 1 ε opt T2
ηtherm.dyn. = 1 −
T2 T4
2
(2.22)
(2.23)
22
Gas turbines
Figure (2.4) shows why there is an optimum pressure ratio in the T-s diagram: both at very large (ε>>εopt) and very small (εεopt opt ε>>
T
ε==ε opt opt 4
3
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